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A continuous [[Function|function]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p120/p120160/p1201601.png" /> on the [[Group|group]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p120/p120160/p1201602.png" /> such that for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p120/p120160/p1201603.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p120/p120160/p1201604.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p120/p120160/p1201605.png" />,
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A continuous [[Function|function]] $f\neq0$ on the [[Group|group]] $G$ such that for all $x_1,\dots,x_n$ in $G$ and $c_1,\dots,c_n\in\mathbf C$,
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p120/p120160/p1201606.png" /></td> </tr></table>
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$$\sum_{i,j}f(x_ix_j^{-1})c_i\overline{c_j}\geq0.$$
  
Examples can be obtained as follows. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p120/p120160/p1201607.png" /> be a [[Unitary representation|unitary representation]] of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p120/p120160/p1201608.png" /> in a [[Hilbert space|Hilbert space]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p120/p120160/p1201609.png" />, and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p120/p120160/p12016010.png" /> be a unit (length) vector. Then
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Examples can be obtained as follows. Let $\pi\colon G\to\Aut(H)$ be a [[Unitary representation|unitary representation]] of $G$ in a [[Hilbert space|Hilbert space]] $H$, and let $u$ be a unit (length) vector. Then
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p120/p120160/p12016011.png" /></td> </tr></table>
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$$f(x)=\langle\pi(x)u,u\rangle$$
  
 
is a positive-definite function.
 
is a positive-definite function.
  
Essentially, these are the only examples. Indeed, there is a [[Bijection|bijection]] between positive-definite functions on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p120/p120160/p12016012.png" /> and isomorphism classes of triples <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p120/p120160/p12016013.png" /> consisting of a unitary representation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p120/p120160/p12016014.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p120/p120160/p12016015.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p120/p120160/p12016016.png" /> and a unit vector <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p120/p120160/p12016017.png" /> that topologically generates <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p120/p120160/p12016018.png" /> under <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p120/p120160/p12016019.png" /> (a [[Cyclic vector|cyclic vector]]). This is the (generalized) Bochner–Herglotz theorem.
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Essentially, these are the only examples. Indeed, there is a [[Bijection|bijection]] between positive-definite functions on $G$ and isomorphism classes of triples $(\pi,H,u)$ consisting of a unitary representation $\pi$ of $G$ on $H$ and a unit vector $u$ that topologically generates $H$ under $\pi(G)$ (a [[Cyclic vector|cyclic vector]]). This is the (generalized) Bochner–Herglotz theorem.
  
See also [[Fourier–Stieltjes transform|Fourier–Stieltjes transform]] (when <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p120/p120160/p12016020.png" />).
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See also [[Fourier–Stieltjes transform|Fourier–Stieltjes transform]] (when $G=\mathbf R$).
  
 
====References====
 
====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  S. Lang,  "<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p120/p120160/p12016021.png" />" , Addison-Wesley  (1975)  pp. Chap. IV, §5</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  G.W. Mackey,  "Unitary group representations in physics, probability and number theory" , Benjamin  (1978)  pp. 147ff</TD></TR></table>
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<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  S. Lang,  "$\SL_2(\mathbf R)$" , Addison-Wesley  (1975)  pp. Chap. IV, §5</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  G.W. Mackey,  "Unitary group representations in physics, probability and number theory" , Benjamin  (1978)  pp. 147ff</TD></TR></table>

Latest revision as of 17:51, 28 November 2018

A continuous function $f\neq0$ on the group $G$ such that for all $x_1,\dots,x_n$ in $G$ and $c_1,\dots,c_n\in\mathbf C$,

$$\sum_{i,j}f(x_ix_j^{-1})c_i\overline{c_j}\geq0.$$

Examples can be obtained as follows. Let $\pi\colon G\to\Aut(H)$ be a unitary representation of $G$ in a Hilbert space $H$, and let $u$ be a unit (length) vector. Then

$$f(x)=\langle\pi(x)u,u\rangle$$

is a positive-definite function.

Essentially, these are the only examples. Indeed, there is a bijection between positive-definite functions on $G$ and isomorphism classes of triples $(\pi,H,u)$ consisting of a unitary representation $\pi$ of $G$ on $H$ and a unit vector $u$ that topologically generates $H$ under $\pi(G)$ (a cyclic vector). This is the (generalized) Bochner–Herglotz theorem.

See also Fourier–Stieltjes transform (when $G=\mathbf R$).

References

[a1] S. Lang, "$\SL_2(\mathbf R)$" , Addison-Wesley (1975) pp. Chap. IV, §5
[a2] G.W. Mackey, "Unitary group representations in physics, probability and number theory" , Benjamin (1978) pp. 147ff
How to Cite This Entry:
Positive-definite function on a group. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Positive-definite_function_on_a_group&oldid=43501
This article was adapted from an original article by M. Hazewinkel (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article